On a question of davenport and lewis and new character sum bounds in finite fields

Abstract

Let χ be a nontrivial multiplicative character of double-struck F signpn. We obtain the following results. (1) Lete ε 0 be given. If B = {Σj=1n xjωj : Xj ∈ [Nj + 1, Nj + Hj] ∩ ℤ j = 1, . . . n} is a box satisfying πj=1n H j > p(2/5+ε)n, then for p > p(ε) we have, denoting χ a nontrivial multiplicative character, |Σ x∈B χ(x)| ≪n p-ε2/4|B| unless n is even, χ is principal on a subfield F2 of size p n/2, and maxξ|B ∩ ξ F2| > p -ε|B|. (2) Assume that A, B ⊂ double-struck F sign p so that |A| > p(4/9)+ε, |B| > p (4/9)+ε, |B + B| < K\B\. Then |Σ x∈A, y∈B χ(x+y)| 1/2 + δ. Then for a nonprincipal multiplicative character χ. | Sigma; x∈I, y∈D χ(x+y)|